Question

A rigid vessel of volume \(0.50 \mathrm{~m}^{3}\) containing \(\mathrm{H}_{2}\) at \(20.5^{\circ} \mathrm{C}\) and a pressure of \(611 \times 10^{3} \mathrm{~Pa}\) is connected to a second rigid vessel of volume \(0.75 \mathrm{~m}^{3}\) containing Ar at \(31.2^{\circ} \mathrm{C}\) at a pressure of \(433 \times 10^{3}\) Pa. A valve separating the two vessels is opened and both are cooled to a temperature of \(14.5^{\circ} \mathrm{C}\). What is the final pressure in the vessels? (a) \(2 \times 10^{5}\) (b) \(3.22 \times 10^{5} \mathrm{~Pa}\) (c) \(4840 \mathrm{~Pa}\) (d) \(4.84 \times 10^{5} \mathrm{~Pa}\)

Short answer

The final pressure is calculated to be approximately \( 3.22 \times 10^{5} \) Pa (Option b).

Step-by-step solution

Convert all temperatures to Kelvin

Convert the given temperatures from Celsius to Kelvin by adding 273.15 to each value. For hydrogen: T1(H2) = 20.5 + 273.15 = 293.65 K. For argon: T1(Ar) = 31.2 + 273.15 = 304.35 K. The final temperature: T2 = 14.5 + 273.15 = 287.65 K.

Calculate the initial number of moles for each gas

Use the ideal gas law, PV = nRT, to solve for the number of moles. For hydrogen, n(H2) = P(V) / (RT). With P(H2) = 611 * 10^3 Pa, V(H2) = 0.50 m^3, and R = 8.314 J/(mol K), n(H2) = (611 * 10^3 * 0.50) / (8.314 * 293.65). For argon, n(Ar) = (433 * 10^3 * 0.75) / (8.314 * 304.35).

Calculate total moles after mixing

After the valve is opened, the gases mix together. The total number of moles, n(total), is the sum of the initial moles of hydrogen and argon: n(total) = n(H2) + n(Ar).

Calculate the final pressure

With the gases mixed and the system cooled to 287.65 K, use the ideal gas law with the combined volume (V(total) = 0.50 m^3 + 0.75 m^3) to find the final pressure, P2. The relationship is P2 = (n(total)RT2) / V(total).

Convert the final pressure to the correct units if necessary

Ensure that the final pressure is in Pascals (Pa) once computed. If it is in another unit, convert it to Pascals.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental principle in chemical thermodynamics that relates the pressure, volume, temperature, and amount of an ideal gas. Expressed as the equation \( PV = nRT \), where \( P \) stands for pressure, \( V \) for volume, \( n \) for moles of the gas, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin. It's essential to convert all temperatures to Kelvin for accurate calculations, as other temperature scales like Celsius won't work within this formula.

When applying the Ideal Gas Law, it's critical to ensure that the units are consistent. Generally, pressure is measured in Pascals (Pa), volume in cubic meters (m^3), and the ideal gas constant \( R \) has a value of 8.314 J/(mol·K) in these units. To use this law for calculating the moles of gas or to find the final or initial pressures and volumes, you'd rearrange the formula to solve for the unknown variable.

In the context of the exercise, this law helps us to determine the initial amount of moles of hydrogen and argon separately before they are mixed, and aids in finding the final pressure after mixing and temperature change.

Gas Mixture Pressure Calculation

Calculating the pressure of a gas mixture involves understanding partial pressures and the law of combining volumes. According to Dalton's Law of Partial Pressures, in a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of individual gases. For a mixture resulting from combining two gas volumes, the total pressure can be calculated if the number of moles, temperature, and combined volume are known.

To find the final pressure of a mixture of gases, you derive the total number of moles from each gas using the Ideal Gas Law, and apply it again with the combined volume and final temperature. In the exercise provided, once the volume of the two gases is confined to a single vessel and allowed to reach equilibrium at a new temperature, the final pressure can be found using the Ideal Gas Law with the updated conditions. This calculation is based on the principle that the combined volume of gas mixtures affects the total pressure when temperature and amount of gas (in moles) are held constant.

Thermodynamic Processes

Thermodynamic processes involve the transfer of energy to or from a system and can affect the state variables like pressure, volume, and temperature. There are different types of processes—such as isobaric (constant pressure), isochoric (constant volume), isothermal (constant temperature), and adiabatic (no heat exchange). In each type of process, certain properties remain constant while others change in response to energy transfer.

In our exercise, we encounter an isochoric process when the gases are initially in rigid vessels and then mixed. Since their volumes don't change, the process is carried out at constant volume. Next, the combined gas mixture undergoes a cooling process to reach a new thermal equilibrium at a lower temperature. Managing these intricate processes involving gas laws, and understanding how they affect each other in a stepwise manner, is paramount to accurately determining the final conditions of the gas mixture.